# A Course in Vertex Algebra by Rosellen M.

By Rosellen M.

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The functor R → (R, o+ ) from vertex Lie algebras to sets with non-negative locality function has a left adjoint (S, o) → R(S, o). Proof. Let F be the free N-fold algebra with derivation generated by the set S. A basis of F is given by B = Bn where B1 := {T k a | a ∈ S, k ≥ 0} and Bn := {ai b | a ∈ Bm , b ∈ Bn−m , 1 ≤ m < n, i ≥ 0} for n ≥ 2. Define a map o : B 2 → N by induction on n: o(T k a, T l b) := o(a, b) + k + l for a, b ∈ S, o(T k a, bi c) := 3 max(o(T k a, b), o(T k a, c), o(b, c)) for a ∈ S, bi c ∈ Bn , and o(ai b, c) := 3 max(o(a, b), o(a, c), o(b, c)) for ai b ∈ Bn , c ∈ B.

This is the Goddard-Kent-Olive coset construction. A homogeneous vector a of a conformal vertex Lie algebra R is primary if Lλ a = (T + Hλ)a. In other words, Ln a = 0 for n ≥ 1. A homogeneous vector a is quasi-primary if L1 a = 0. For example, L is quasi-primary and primary iff cˆL = 0. For a primary, T a is primary iff ha = 0. Central elements are primary. Many conformal vertex Lie algebras are generated by primary vectors. A conformal vertex Lie subalgebra of R is a graded vertex Lie subalgebra R′ together with a quasi-primary conformal vector L′ ∈ R′ .

20 2 Vertex Lie Algebras a(w)i b(w) = (−1)j i j [a(i−j) , b(t+j) ] w−t−1 with [a(i−j) , b(t+j) ] ∈ gha +hb −i−1−t−1 . We have (∂z a(z))n = (∂z a(z))(n+ha ) = −(n + ha )an . The weak commutator formula for the an is N [an , bm ] = i=0 n + ha − 1 i cn+m i since (n + ha − 1) + (m + hb − 1) − i = n + m + (ha + hb − i − 1) − 1. Let E be a vector space. A distribution a(z) ∈ End(E)[[z ±1 ]] is a field on E if a(z)b ∈ E((z)) for any b ∈ E. 1 we discuss this notion in detail. Here we only remark that the space Endv (E) of fields is an unbounded cont t−i formal subalgebra of gl(E)[[z ±1 ]] since a(w)t b(w) = .